3 Answers
3

Barbarians (and Assassins, in the expansion) are the only classes who can dual wield.

Barbarians can dual wield any one-handed melee weapons - one-handed swords, one-handed axes, one-handed hammers. They can also dual wield throwing weapons like javelins, but without the Double Throw skill they can't throw from both hands. As a final point, they can also wield a two-handed sword in a single hand in order to dual-wield. This does reduce the damage, and the power of the weapon for both single-handed and two-handed version is listed on hovering. Assassins can only dual wield their claw weapons.

When you perform normal attacks, you will alternate between the weapons in your hands. The attack speed is calculated for each arm swing, so different weapon speeds will result in different attack speeds. However, the speed is generally faster than if you just swung either of the weapons twice. Clicking faster has no effect, and the damage is simply that of each weapon hitting. And of course, dual wielding is necessary to use a number of combat skills like Double Swing, Double Throw, and Frenzy for the Barbarian, and some martial arts skills for the Assassin.

Your right hand strikes first, but in most scenarios it's fairly moot whether you keep your weapons on one hand or the other.

Slight correction: Without Double Throw, Barbarians can't throw from both hands simultaneously. But you can set the left-click skill to one hand and the right-click skill to the other hand if you want, so you can throw from either hand without having to switch.
–
KyralessaAug 15 '10 at 1:55

2

There ARE some important things to note about which hands to use which weapon. Keep in mind that the whirlwind speed is determined by the primary hand's weapon speed, this is especially important if you are trying to hit a specific frame speed for WW on an assassin when selecting the right type of claw to use. There are ways you can force the game to use the offhand claw to calculate WW speed, but you will have to do that upon every death.
–
z 'Feb 8 '11 at 1:18

Every Assassin should put at least one skill point into Weapon Block, and therefore should be dual wielding by level 12. Everyone I know (including me) that's played assassin has developed the Shadow Discipline tree at least to Weapon Block.
–
MBraedleyNov 10 '11 at 0:45

Assassins can only dual wield claws.
Barbarians can dual wield any two 1-handed weapons. Also all swords can be used as a 1-handed weapon on a barbarian. For example, you can dual wield Patriarch sword (2-hand) and Grandfather sword (2-hand). But you can't dual wield with Axe of Fechmar because it is a 2-handed axe not a 2-handed sword.

There is a complicated answer to this and I can only provide partial information. See elsewhere for subtleties of left and right hand weapons; but generally the right-hand weapon is above the glove, the left-hand is above the boot.

If you are using normal attack (non-assassin), only your right-hand weapon will be used

If you are using normal attack (assassin), the claws alternate

If you are using double swing, double throw, or frenzy, both will be used per click.

If you are using whirlwind, the right-hand weapon will always hit first. Then for the rest of the attack, both weapons will hit at the same time.

See #3.

For kicking (dragon talon), trap laying, double swing, double throw, frenzy, and whirlwind, the base weapon speeds of the weapons are first averaged. For example phaseblade has -30 base speed and a berserker axe has 0 base speed; their average is -15. Next the IAS (increased attack speed) of the right-hand weapon will be used to calculate your final speed. The IAS from the left-hand weapon will be completely ignored.

For more details, check out notable sites like incgamer.com or Blizzard's website for Diablo 2.